On the subgroup perfect codes in Cayley graphs

Abstract

Let $\Gamma =\mathrm {Cay}(G,S)$ be a Cayley graph on a finite group G. A perfect code in $\Gamma$ is a subset C of G such that every vertex in $G\setminus C$ is adjacent to exactly one vertex in C and vertices of C are not adjacent to each other. In Zhang and Zhou (Eur J Comb 91:103228, 2021) it is proved that if H is a subgroup of G whose Sylow 2-subgroup is a perfect code of G, then H is a perfect code of G. Also they proved that if G is a metabelian group and H is a normal subgroup of G, then H is a perfect code of G if and only if a Sylow 2-subgroup of H is a perfect code of G. As a generalization, we prove that this result holds for each finite group G. Also they proved that if G is a nilpotent group and H is a subgroup of G, then H is a perfect code of G if and only if the Sylow 2-subgroup of H is a perfect code of G. We generalize this result by proving that the same result holds for every group with a normal Sylow 2-subgroup. In the rest of the paper, we classify groups whose set of all subgroup perfect codes forms a chain of subgroups and also we determine groups with exactly two proper non-trivial subgroup perfect codes.